It is known that the image of Santa Claus came from Saint Nicholas (of Myra), which to this day is said to visit kids on Dec 6th to bring small gifts.

The myth developed as a mix of Saint Nicholas stories, Roman Saturn and Norse Odin.

In some european parts, the role of the gift-bringer was replaced with baby jesus (which is also false, since the gifts were brought by the 3 wise men on Jan 6th... babies generally don't gift anything). Until recently it was quite common, for example in spain, to give gifts on jan 6th, not on christmas.

With colonization, the Dutch version of "Sinterklaas" was spread to the colonies.

The first appearance of Santa flying a wagon was in 1809 by Washington Irving. In 1822 a Poem by Clement Clark Moore went viral, spreading the idea further.

Coca Cola just took those mostly regional stories and put them on nation-wide stage first and world-wide stage later, spreading the idea around the world and establishing it further.

I'm pretty sure that some people learned about the stories from the coca cola commercials, but they merely adopted it and gave it a bit of their own spin (like santa wanting you to put out a coke for him instead of milk and cookies)